# theorem

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## theorem,

in mathematics and logic, statement in words or symbols that can be established by means of deductive logic; it differs from an axiomaxiom,
in mathematics and logic, general statement accepted without proof as the basis for logically deducing other statements (theorems). Examples of axioms used widely in mathematics are those related to equality (e.g.
in that a proofproof,
in mathematics, finite sequence of propositions each of which is either an axiom or follows from preceding propositions by one of the rules of logical inference (see symbolic logic).
is required for its acceptance. A lemma is a theorem that is demonstrated as an intermediate step in the proof of another, more basic theorem. A corollary is a theorem that follows as a direct consequence of another theorem or an axiom. There are many famous theorems in mathematics, often known by the name of their discoverer, e.g., the Pythagorean Theorem, concerning right triangles. One of the most famous problems of number theory was the proof of Fermat's Last Theorem (see Fermat, Pierre deFermat, Pierre de
, 1601–65, French mathematician. A magistrate whose avocation was mathematics, Fermat is known as a founder of modern number theory and probability theory.
); the theorem states that for an integer n greater than 2 the equation xn+yn=zn admits no solutions where x, y, and z are also integers.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## Theorem

a statement, in some deductive theory, that has been or is to be proved (seeDEDUCTION). Examples of deductive theories are provided by mathematics, logic, theoretical mechanics, and some branches of physics. Every such theory consists of theorems that are proved one after another on the basis of previously proved theorems. The first statements in the deductive process are accepted without proof and thus form the logical basis of the given area of the theory. Such primitive statements are called axioms.

In the formulation of a theorem, a distinction is made between the hypothesis and the conclusion. Consider, for example, the following two theorems: (1) If the sum of the digits in a number is divisible by 3, then the number is itself divisible by 3. (2) If one of the angles in a triangle is a right angle, then the other two angles are acute. In these examples the word “if” is followed by the hypothesis of the theorem, and the word “then” is followed by the conclusion. Every theorem can be expressed in this form. For example, the theorem “Any angle inscribed in a semicircle is a right angle” can be expressed “If an angle is inscribed in a semicircle, then the angle is a right angle.”

The converse of a theorem expressed in the form “If..., then...” is obtained by interchanging the hypothesis and the conclusion. A theorem and its converse are converses of each other. In general, the validity of a theorem does not imply the validity of its converse. For example, the converse of theorem (1) is true, but the converse of theorem (2) is false. If a theorem and its converse are both true, then the hypothesis of either theorem is a necessary and sufficient condition for the validity of the conclusion (seeNECESSARY AND SUFFICIENT CONDITIONS).

If the hypothesis and conclusion of a theorem are replaced by their negations, then the inverse of the given theorem is obtained. The inverse of a theorem is equivalent to the theorem’s converse. Moreover, the converse of the inverse of a theorem is equivalent to the original theorem. Consequently, the validity of a theorem can be demonstrated by both a direct and an indirect proof. An indirect proof, also known as reductio ad absurdum, involves showing that the negation of the hypothesis of the theorem follows from the negation of the theorem’s conclusion. This method of proof is very widely used in mathematics.

## theorem

[′thir·əm]
(mathematics)
A proven mathematical statement.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.

## theorem

Maths Logic a statement or formula that can be deduced from the axioms of a formal system by means of its rules of inference
Collins Discovery Encyclopedia, 1st edition © HarperCollins Publishers 2005
References in periodicals archive ?
By Theorems 18 and 20 we have for [lambda] [member of] [C.sub.+], [w.sub.[phi]] a.e.
(1) Theorems 15 and 17 can be obtained from Theorems 9 and 12, respectively.
Lopez, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order, 22 (2005), 223-239.
In [1], Florin Ambro introduced the notion of quasi-log varieties, which are now called quasi-log schemes, in order to establish the cone and contraction theorem for generalized log varieties.
Table 1 suggests that our Theorem 11 gives significantly better bounds than those obtained from any result, including Theorems 5 and 7.
Changing the order of the integration valid under the condition given with the theorem, we obtain
The proof of this theorem in [7] is nontrivial and uses the deep and powerful Kolmogorov Superposition Theorem which answered Hilbert's 13th Problem.
In [12], theorems on universality of F ([zeta](s, [[alpha].sub.1]), ..., [zeta](s, [[alpha].sub.n])) for some other classes of operators F also can be found.
In this work, considering the target space Y = M a manifold and H a proper nontrivial subgroup of G, we prove the following formulation of the BorsukUlam theorem for manifolds in terms of (H, G)-coincidence.
Applications: The following are only a few examples of abstracts of articles containing some important fixed point theorems on convex-valued multimaps:
Carefully analyzing the analytical formulas for the solutions of system (1) given in Theorems 4-9, it is possible (under various assumptions for the roots of (6) and for the initial data) to derive asymptotic formulas for the solutions of system (1) or simplify the formulas for the exact solutions.
Start with the one-dimensional theorem where a line length x inline with a line length y results in a line length z, inscribed in a circle with diameter z (Figure 3a).

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